COMPLETELY MONOTONIC FUNCTIONS RELATED TO LOGARITHMIC DERIVATIVES OF ENTIRE FUNCTIONS

2011 ◽  
Vol 09 (04) ◽  
pp. 409-419 ◽  
Author(s):  
HENRIK L. PEDERSEN
Keyword(s):  
Entire Functions ◽  
Logarithmic Derivative ◽  
Psi Function ◽  
Stieltjes Function ◽  
Gamma Functions ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions ◽  
Derivatives Of ◽  
Logarithmic Derivatives

The logarithmic derivative l(x) of an entire function of genus p and having only non-positive zeros is represented in terms of a Stieltjes function. As a consequence, (-1)p(xml(x))(m+p) is a completely monotonic function for all m ≥ 0. This generalizes earlier results on complete monotonicity of functions related to Euler's psi-function. Applications to Barnes' multiple gamma functions are given.

scholarly journals A class of completely monotonic functions involving the polygamma functions

2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
Li-Chun Liang ◽  
Li-Fei Zheng ◽  
Aying Wan
Keyword(s):  
Gamma Function ◽  
Sufficient Conditions ◽  
Psi Function ◽  
Gamma Functions ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Polygamma Functions ◽  
Completely Monotonic Functions ◽  
Alpha Beta ◽  
Necessary And Sufficient

AbstractLet $\Gamma (x)$ Γ ( x ) denote the classical Euler gamma function. We set $\psi _{n}(x)=(-1)^{n-1}\psi ^{(n)}(x)$ ψ n ( x ) = ( − 1 ) n − 1 ψ ( n ) ( x ) ($n\in \mathbb{N}$ n ∈ N ), where $\psi ^{(n)}(x)$ ψ ( n ) ( x ) denotes the nth derivative of the psi function $\psi (x)=\Gamma '(x)/\Gamma (x)$ ψ ( x ) = Γ ′ ( x ) / Γ ( x ) . For λ, α, $\beta \in \mathbb{R}$ β ∈ R and $m,n\in \mathbb{N}$ m , n ∈ N , we establish necessary and sufficient conditions for the functions $$ L(x;\lambda ,\alpha ,\beta )=\psi _{m+n}(x)-\lambda \psi _{m}(x+ \alpha ) \psi _{n}(x+\beta ) $$ L ( x ; λ , α , β ) = ψ m + n ( x ) − λ ψ m ( x + α ) ψ n ( x + β ) and $-L(x;\lambda ,\alpha ,\beta )$ − L ( x ; λ , α , β ) to be completely monotonic on $(-\min (\alpha ,\beta ,0),\infty )$ ( − min ( α , β , 0 ) , ∞ ) .As a result, we generalize and refine some inequalities involving the polygamma functions and also give some inequalities in terms of the ratio of gamma functions.

scholarly journals Monotonicity and complete monotonicity of two functions constituted via three derivatives of a function involving trigamma function

2020 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Sufficient Conditions ◽  
Laplace Transforms ◽  
Necessary And Sufficient Conditions ◽  
Complete Monotonicity ◽  
Convolution Theorem ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions ◽  
Necessary And Sufficient ◽  
Derivatives Of

In the paper, by convolution theorem of the Laplace transforms, a monotonicity rule for the ratio of two Laplace transforms, Bernstein's theorem for completely monotonic functions, and other analytic techniques, the author (1) presents the decreasing monotonicity of a ratio constituted via three derivatives of a function involving trigamma function; (2) discovers necessary and sufficient conditions for a function constituted via three derivatives of a function involving trigamma function to be completely monotonic. These results conform previous guesses posed by the author.

INEQUALITIES ASSOCIATED WITH RATIOS OF GAMMA FUNCTIONS

2018 ◽  
Vol 97 (3) ◽  
pp. 453-458
Author(s):  
JENICA CRINGANU
Keyword(s):  
Gamma Function ◽  
Monotonic Function ◽  
Completely Monotonic Function ◽  
Gamma Functions ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions ◽  
Appl Math

We use properties of the gamma function to estimate the products$\prod _{k=1}^{n}(4k-3)/4k$and$\prod _{k=1}^{n}(4k-1)/4k$, motivated by the work of Chen and Qi [‘Completely monotonic function associated with the gamma function and proof of Wallis’ inequality’,Tamkang J. Math.36(4) (2005), 303–307] and Morticiet al.[‘Completely monotonic functions and inequalities associated to some ratio of gamma function’,Appl. Math. Comput.240(2014), 168–174].

scholarly journals Complete monotonicity of a difference constituted by four derivatives of a function involving trigamma function

2020 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Sufficient Conditions ◽  
Laplace Transforms ◽  
Necessary And Sufficient Conditions ◽  
Complete Monotonicity ◽  
Convolution Theorem ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions ◽  
Necessary And Sufficient ◽  
Derivatives Of

In the paper, by virtue of convolution theorem for the Laplace transforms, Bernstein's theorem for completely monotonic functions, and other techniques, the author finds necessary and sufficient conditions for a difference constituted by four derivatives of a function involving trigamma function to be completely monotonic.

scholarly journals On Some Complete Monotonicity of Functions Related to Generalized k − Gamma Function

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Hesham Moustafa ◽  
Hanan Almuashi ◽  
Mansour Mahmoud
Keyword(s):  
Lower Bounds ◽  
Gamma Function ◽  
Logarithmic Derivative ◽  
Complete Monotonicity ◽  
Upper And Lower Bounds ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions

In this paper, we presented two completely monotonic functions involving the generalized k − gamma function Γ k x and its logarithmic derivative ψ k x , and established some upper and lower bounds for Γ k x in terms of ψ k x .

scholarly journals Necessary and sufficient conditions for complete monotonicity and monotonicity of two functions defined by two derivatives of a function involving trigamma function

2021 ◽  
pp. 14-14
Author(s):  
Feng Qi
Keyword(s):  
Exponential Function ◽  
Sufficient Conditions ◽  
Laplace Transforms ◽  
Necessary And Sufficient Conditions ◽  
Complete Monotonicity ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions ◽  
Necessary And Sufficient ◽  
Derivatives Of

In the paper, by virtue of convolution theorem for the Laplace transforms, Bernstein's theorem for completely monotonic functions, some properties of a function involving exponential function, and other analytic techniques, the author finds necessary and sufficient conditions for two functions defined by two derivatives of a function involving trigamma function to be completely monotonic or monotonic. These results generalize corresponding known ones.

Completely monotonicity of class functions involving the polygamma and related functions

2020 ◽  
pp. 2150064
Author(s):  
B. Ravi ◽  
A. Venakata Lakshmi
Keyword(s):  
Lower Bound ◽  
Zeta Function ◽  
Complete Monotonicity ◽  
Open Problems ◽  
Gamma Functions ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Polygamma Functions ◽  
Completely Monotonic Functions ◽  
Monotonic Properties

In this paper, the authors prove some inequalities and completely monotonic properties of polygamma functions. As an application, we give lower bound for the zeta function on natural numbers. Partially, we answer the fifth and sixth open problems listed in [F. Qi and R. P. Agarwal, On complete monotonicity for several classes of functions related to ratios of gamma functions, J. Inequalities Appl. 2019(36) (2019) 42]. We propose two open problems on completely monotonic functions related to polygamma functions.

scholarly journals Complete monotonicity of a difference defined by four derivatives of a function containing trigamma function

2020 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Gamma Function ◽  
Integral Inequality ◽  
Sufficient Conditions ◽  
Complete Monotonicity ◽  
Logarithmic Convexity ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions ◽  
Necessary And Sufficient ◽  
Derivatives Of

In the paper, by virtue of convolution theorem for the Laplace transforms, logarithmic convexity of the gamma function, Bernstein's theorem for completely monotonic functions, and other techniques, the author finds necessary and sufficient conditions for a difference defined by four derivatives of a function containing trigamma function to be completely monotonic. Moreover, by virtue of Cebysev integral inequality, the author presents logarithmic convexity of the sequence of polygamma functions.

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